SYSTEMS OF PARTICLES AND ROTATIONAL MOTION

1 CHAPTER SEVEN. SYSTEMS OF PARTICLES AND ROTATIONAL MOTION . INTRODUCTION. In the earlier chapters we primarily considered the MOTION of a single particle. (A particle is represented as a point mass. Introduction It has practically no size.) We applied the results of our Centre of mass study even to the MOTION of bodies of finite size, assuming MOTION of centre of mass that MOTION of such bodies can be described in terms of the Linear momentum of a MOTION of a particle. system of PARTICLES Any real body which we encounter in daily life has a Vector product of two finite size.

2 In dealing with the MOTION of extended bodies vectors (bodies of finite size) often the idealised model of a particle is Angular velocity and its inadequate. In this chapter we shall try to go beyond this relation with linear velocity inadequacy. We shall attempt to build an understanding of T or que and angular the MOTION of extended bodies. An extended body, in the momentum first place, is a system of PARTICLES . We shall begin with the Equilibrium of a rigid body consideration of MOTION of the system as a whole. The centre Moment of inertia of mass of a system of PARTICLES will be a key concept here.

3 Theorems of perpendicular We shall discuss the MOTION of the centre of mass of a system and parallel axes of PARTICLES and usefulness of this concept in understanding Kinematics of ROTATIONAL the MOTION of extended bodies. MOTION about a fixed axis A large class of problems with extended bodies can be Dynamics of r otational solved by considering them to be rigid bodies. Ideally a MOTION about a fixed axis rigid body is a body with a perfectly definite and Angular momentum in case unchanging shape. The distances between all pairs of of rotation about a fixed axis PARTICLES of such a body do not change.

4 It is evident from Rolling MOTION this definition of a rigid body that no real body is truly rigid, since real bodies deform under the influence of forces. But in Summary many situations the deformations are negligible. In a number Points to Ponder of situations involving bodies such as wheels, tops, steel Exercises beams, molecules and planets on the other hand, we can ignore Additional exercises that they warp, bend or vibrate and treat them as rigid. What kind of MOTION can a rigid body have? Let us try to explore this question by taking some examples of the MOTION of rigid bodies.

5 Let us begin with a rectangular block sliding down an inclined plane without any sidewise 142 PHYSICS. that it does not have translational MOTION is to fix it along a straight line. The only possible MOTION of such a rigid body is rotation. The line along which the body is fixed is termed as its axis of rotation. If you look around, you will come across many examples of rotation about an axis, a ceiling fan, a potter's wheel, a giant wheel in a fair, a merry-go-round and so on (Fig (a) and (b)). Fig Translational (sliding) MOTION of a block down an inclined plane.

6 (Any point like P1 or P2 of the block moves with the same velocity at any instant of time.). movement. The block is a rigid body. Its MOTION down the plane is such that all the PARTICLES of the body are moving together, they have the same velocity at any instant of time. The rigid body here is in pure translational MOTION (Fig. ). In pure translational MOTION at any instant of time all PARTICLES of the body have the same velocity. Consider now the rolling MOTION of a solid metallic or wooden cylinder down the same (a). inclined plane (Fig.)

7 The rigid body in this problem, namely the cylinder, shifts from the top to the bottom of the inclined plane, and thus, has translational MOTION . But as Fig. shows, all its PARTICLES are not moving with the same velocity at any instant. The body therefore, is not in pure translation. Its MOTION is translation plus something else.'. Fig. Rolling MOTION of a cylinder It is not pure translational MOTION . Points P1, P2, P3 and P4. have different velocities (shown by arrows). (b). at any instant of time. In fact, the velocity of Fig.

8 Rotation about a fixed axis the point of contact P3 is zero at any instant, (a) A ceiling fan if the cylinder rolls without slipping. (b) A potter's wheel. In order to understand what this something else' is, let us take a rigid body so constrained Let us try to understand what rotation is, that it cannot have translational MOTION . The what characterises rotation. You may notice most common way to constrain a rigid body so that in rotation of a rigid body about a fixed SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 143. Fig. (a) A spinning top (The point of contact of the top with the ground, its tip O, is fixed.)

9 Fig. A rigid body rotation about the z-axis (Each point of the body such as P 1 or P2 describes a circle with its centre (C1. or C2) on the axis. The radius of the circle ( r 1 or r 2 ) is the perpendicular distance of the point (P1 or P2 ) from the axis. A point on the axis like P3 remains stationary). axis, every particle of the body moves in a circle, which lies in a plane perpendicular to Fig. (b) An oscillating table fan. The pivot of the the axis and has its centre on the axis. Fig. fan, point O, is fixed. shows the ROTATIONAL MOTION of a rigid body about a fixed axis (the z-axis of the frame of In some examples of rotation, however, the reference).

10 Let P1 be a particle of the rigid body, axis may not be fixed. A prominent example of arbitrarily chosen and at a distance r1 from fixed this kind of rotation is a top spinning in place axis. The particle P1 describes a circle of radius [Fig. (a)]. (We assume that the top does not r1 with its centre C1 on the fixed axis. The circle slip from place to place and so does not have lies in a plane perpendicular to the axis. The translational MOTION .) We know from experience figure also shows another particle P2 of the rigid that the axis of such a spinning top moves body, P2 is at a distance r2 from the fixed axis.